Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 18 min ago
PreprintCC BY 4.0arXiv

Matroids with cycle systems are regular

  • Anton Dochtermann
  • Evan Huang
  • Kai Mawhinney
  • Suho Oh
  • Yuchen Xu
Open at sourcePDFDOICite

Abstract

A cycle system for a matroid M is a collection of cycles (unions of circuits) whose intersection properties mimic the cut sets of a graph. Cycle systems were introduced by Corry, the first author, McClain, Perkinson, and Yi, who showed that the h-vector of any matroid that admits a cycle system is a pure O-sequence, confirming a conjecture of Stanley for this class. Those authors also conjectured that any matroid admitting a cycle system must be binary. Here we answer this conjecture in the affirmative, and prove the stronger result that any such matroid must in fact be regular (representable over any field). From this, we conclude that if M is connected, every cycle system for M is a basis for its circuit space of. We establish other properties of cycle systems along the way, which may be of independent interest.

Related

  1. Nonnegative conorms, regular matroids, and the tropical Schottky problem
    Yelena Mandelshtam
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  2. Freeness of Arrangements with Regular Underlying Matroids
    Weikang Liang, Suijie Wang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. Logarithmic basis number of graphs and regular matroids
    Kolja Knauer
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. Construction of Regular Fat Linear Sets and Non-Representable q-Matroids
    Yufei Qiu, Valentino Smaldore, Zhenpeng Wang +2
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Almost entropic matroids and (k,L)-systems
    Andrey Kupavskii, Nikolai Terekhov
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{dochtermann2026matroids,
  title = {{Matroids with cycle systems are regular}},
  author = {Anton Dochtermann and Evan Huang and Kai Mawhinney and Suho Oh and Yuchen Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13361},
  eprint = {2609.13361},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13361},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Matroids with cycle systems are regular
AU  - Anton Dochtermann
AU  - Evan Huang
AU  - Kai Mawhinney
AU  - Suho Oh
AU  - Yuchen Xu
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.13361
UR  - https://arxiv.org/abs/2609.13361
AB  - A cycle system for a matroid M is a collection of cycles (unions of circuits) whose intersection properties mimic the cut sets of a graph. Cycle systems were introduced by Corry, the first author, McClain, Perkinson, and Yi, who showed that the h-vector of any matroid that admits a cycle system is a pure O-sequence, confirming a conjecture of Stanley for this class. Those authors also conjectured that any matroid admitting a cycle system must be binary. Here we answer this conjecture in the affirmative, and prove the stronger result that any such matroid must in fact be regular (representable over any field). From this, we conclude that if M is connected, every cycle system for M is a basis for its circuit space of. We establish other properties of cycle systems along the way, which may be of independent interest.
ER  -